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    Scott's theorem shows measurability implies V≠L, but this... — Carmelics
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    Challenges→L cannot recognize that κ is a measurable cardinal

    Scott's theorem shows measurability implies V≠L, but this is an external judgment requiring a richer metatheory, not a failure of L's internal recognition.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.L's internal logic is consistent; external theorems about L don't contradict L's self-conception, only our external knowledge.
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    • 2.Metatheory can access facts about models that the model itself cannot recognize without additional axioms beyond its own theory.
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    • 3.Scott's theorem uses strong assumptions (measurable cardinals) absent from L; the gap reflects mathematical expressiveness, not L's failure.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If V≠L is true, L's claim to capture all sets is objectively false, regardless of whether L 'recognizes' this internally.
      ?

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    • 2.The distinction between 'internal recognition' and 'external judgment' dissolves if both operate within the same mathematical reality.
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    • 3.Requiring a richer metatheory to expose L's inadequacy suggests L itself is inadequate—the metatheory isn't external but foundational.
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    Connections

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    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    If V≠L is true, L's claim to capture all sets is objectively false, regardless o...L cannot recognize that κ is a measurable cardinalL's internal logic is consistent; external theorems about L don't contradict L's...Metatheory can access facts about models that the model itself cannot recognize ...
    +3 moreShow less
    Requiring a richer metatheory to expose L's inadequacy suggests L itself is inad...Scott's theorem uses strong assumptions (measurable cardinals) absent from L; th...The distinction between 'internal recognition' and 'external judgment' dissolves...

    Details

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    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit